After substituting the equations of motion of auxiliary fields in \(\hat{\cal {L}}\) , and comparing the Lagrangians (i.e. \(\hat{\cal {L}}\)  and \(\hat{\cal {L}^{\prime }}\) ), we find that we must take \(\beta _3=\gamma _2=-{\textstyle {\frac{\scriptstyle 1}{\scriptstyle 2} } }\) .
Thus, to summarise, we find that the BRST formulation in [1]}
corresponds to the specialisation of our general formulation in which
\(\beta _1&=& \xi _B\, \xi _d\,,\qquad \beta _2=-\xi _B\,,\qquad \beta _3=-{\textstyle {\frac{\scriptstyle 1}{\scriptstyle 2} } }\,,\qquad \beta _4=-\bar{\beta }_4=2\beta _5\,,\qquad \beta _6=0\,,\nonumber \\\gamma _1 &=& -\xi _h\,,\qquad \gamma _2=-{\textstyle {\frac{\scriptstyle 1}{\scriptstyle 2} } }\,,\qquad \gamma _3={\textstyle {\frac{\scriptstyle 1}{\scriptstyle 4} } }\,.\) 
